Pub Date : 2011-10-31DOI: 10.1080/15427951.2012.625246
A. Bonato, J. Janssen, P. Prałat
Abstract We study the link structure of online social networks (OSNs) and introduce a new model for such networks that may help in inferring their hidden underlying reality. In the geo-protean (GEO-P) model for OSNs, nodes are identified with points in Euclidean space, and edges are stochastically generated by a mixture of the relative distance of nodes and a ranking function. With high probability, the GEO-P model generates graphs satisfying many observed properties of OSNs, such as power-law degree distributions, the small-world property, densification power law, and bad spectral expansion. We introduce the dimension of an OSN based on our model and examine this new parameter using actual OSN data. We discuss how the geo-protean model may eventually be used as a tool to group users with similar attributes using only the link structure of the network.
{"title":"Geometric Protean Graphs","authors":"A. Bonato, J. Janssen, P. Prałat","doi":"10.1080/15427951.2012.625246","DOIUrl":"https://doi.org/10.1080/15427951.2012.625246","url":null,"abstract":"Abstract We study the link structure of online social networks (OSNs) and introduce a new model for such networks that may help in inferring their hidden underlying reality. In the geo-protean (GEO-P) model for OSNs, nodes are identified with points in Euclidean space, and edges are stochastically generated by a mixture of the relative distance of nodes and a ranking function. With high probability, the GEO-P model generates graphs satisfying many observed properties of OSNs, such as power-law degree distributions, the small-world property, densification power law, and bad spectral expansion. We introduce the dimension of an OSN based on our model and examine this new parameter using actual OSN data. We discuss how the geo-protean model may eventually be used as a tool to group users with similar attributes using only the link structure of the network.","PeriodicalId":38105,"journal":{"name":"Internet Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2011-10-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1080/15427951.2012.625246","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"59946389","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2011-10-12DOI: 10.1080/15427951.2013.836581
Charalampos E. Tsourakakis
Abstract Vertex similarity is a major concept in network science with a wide range of applications. In this work we provide novel perspectives on finding (dis)similar vertices within a network and across two networks with the same number of vertices (graph matching). With respect to the former problem, we propose to optimize a geometric objective that allows us to express each vertex uniquely as a convex combination of a few extreme types of vertices. Our method has the important advantage of supporting efficiently several types of queries such as, which other vertices are most similar to this vertex? by using appropriate data structures and by mining interesting patterns in the network. With respect to the latter problem (graph matching) we propose the generalized condition number—a quantity widely used in numerical analysis— κ(LG, LH) of the Laplacian matrix representations of G, H as a measure of graph similarity, where G, H are the graphs of interest. We show that this objective has a solid theoretical basis, and, we propose a deterministic and a randomized graph alignment algorithm. We evaluate our algorithms on both synthetic and real data. We observe that our proposed methods achieve high-quality results and provide us with significant insights into the network structure.
{"title":"Toward Quantifying Vertex Similarity in Networks","authors":"Charalampos E. Tsourakakis","doi":"10.1080/15427951.2013.836581","DOIUrl":"https://doi.org/10.1080/15427951.2013.836581","url":null,"abstract":"Abstract Vertex similarity is a major concept in network science with a wide range of applications. In this work we provide novel perspectives on finding (dis)similar vertices within a network and across two networks with the same number of vertices (graph matching). With respect to the former problem, we propose to optimize a geometric objective that allows us to express each vertex uniquely as a convex combination of a few extreme types of vertices. Our method has the important advantage of supporting efficiently several types of queries such as, which other vertices are most similar to this vertex? by using appropriate data structures and by mining interesting patterns in the network. With respect to the latter problem (graph matching) we propose the generalized condition number—a quantity widely used in numerical analysis— κ(LG, LH) of the Laplacian matrix representations of G, H as a measure of graph similarity, where G, H are the graphs of interest. We show that this objective has a solid theoretical basis, and, we propose a deterministic and a randomized graph alignment algorithm. We evaluate our algorithms on both synthetic and real data. We observe that our proposed methods achieve high-quality results and provide us with significant insights into the network structure.","PeriodicalId":38105,"journal":{"name":"Internet Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2011-10-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1080/15427951.2013.836581","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"59947784","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2011-08-30DOI: 10.1080/15427951.2011.601233
E. Jonckheere, P. Lohsoonthorn, F. Ariaei
Abstract In this paper, we extend the concept of scaled Gromov hyperbolic graph, originally developed for the thin triangle condition (TTC), to the computationally simplified, but less intuitive, four-point condition (FPC). The original motivation was that for a large but finite network graph to enjoy some of the typical properties to be expected in negatively curved Riemannian manifolds, the delta measuring the thinness of a triangle scaled by its diameter must be below a certain threshold all across the graph. Here we develop various ways of scaling the 4-point delta, and develop upper bounds for the scaled 4-point delta in various spaces. A significant theoretical advantage of the TTC over the FPC is that the latter allows for a Gromov-like characterization of Ptolemaic spaces. As a major network application, it is shown that scale-free networks tend to be scaled Gromov hyperbolic, while small-world networks are rather scaled positively curved.
{"title":"Scaled Gromov Four-Point Condition for Network Graph Curvature Computation","authors":"E. Jonckheere, P. Lohsoonthorn, F. Ariaei","doi":"10.1080/15427951.2011.601233","DOIUrl":"https://doi.org/10.1080/15427951.2011.601233","url":null,"abstract":"Abstract In this paper, we extend the concept of scaled Gromov hyperbolic graph, originally developed for the thin triangle condition (TTC), to the computationally simplified, but less intuitive, four-point condition (FPC). The original motivation was that for a large but finite network graph to enjoy some of the typical properties to be expected in negatively curved Riemannian manifolds, the delta measuring the thinness of a triangle scaled by its diameter must be below a certain threshold all across the graph. Here we develop various ways of scaling the 4-point delta, and develop upper bounds for the scaled 4-point delta in various spaces. A significant theoretical advantage of the TTC over the FPC is that the latter allows for a Gromov-like characterization of Ptolemaic spaces. As a major network application, it is shown that scale-free networks tend to be scaled Gromov hyperbolic, while small-world networks are rather scaled positively curved.","PeriodicalId":38105,"journal":{"name":"Internet Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2011-08-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1080/15427951.2011.601233","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"59946603","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2011-08-19DOI: 10.1080/15427951.2011.646176
E. Grechnikov
Abstract In this paper, we study some important statistics of the random graph H (t) a,k in the Buckley–Osthus model, where t is the number of nodes, kt is the number of edges (so that ), and a>0 is the so-called initial attractiveness of a node. This model is a modification of the well-known Bollobás–Riordan model. First, we find a new asymptotic formula for the expectation of the number R(d, t) of nodes of a given degree d in a graph in this model. Such a formula is known for and d⩽t 1/100(a+1). Both restrictions are unsatisfactory from theoretical and practical points of view. We completely remove them. Then we calculate the covariances between any two quantities R(d 1, t) and R(d 2, t), and using the second-moment method we show that R(d, t) is tightly concentrated around its mean for all possible values of d and t. Furthermore, we study a more complicated statistic of the web graph: X(d 1, d 2, t) is the total number of edges between nodes whose degrees are equal to d 1 and d 2 respectively. We also find an asymptotic formula for the expectation of X(d 1, d 2, t) and prove a tight concentration result. Again, we do not impose any substantial restrictions on the values of d 1, d 2, and t.
{"title":"Degree Distribution and Number of Edges between Nodes of Given Degrees in the Buckley–Osthus Model of a Random Web Graph","authors":"E. Grechnikov","doi":"10.1080/15427951.2011.646176","DOIUrl":"https://doi.org/10.1080/15427951.2011.646176","url":null,"abstract":"Abstract In this paper, we study some important statistics of the random graph H (t) a,k in the Buckley–Osthus model, where t is the number of nodes, kt is the number of edges (so that ), and a>0 is the so-called initial attractiveness of a node. This model is a modification of the well-known Bollobás–Riordan model. First, we find a new asymptotic formula for the expectation of the number R(d, t) of nodes of a given degree d in a graph in this model. Such a formula is known for and d⩽t 1/100(a+1). Both restrictions are unsatisfactory from theoretical and practical points of view. We completely remove them. Then we calculate the covariances between any two quantities R(d 1, t) and R(d 2, t), and using the second-moment method we show that R(d, t) is tightly concentrated around its mean for all possible values of d and t. Furthermore, we study a more complicated statistic of the web graph: X(d 1, d 2, t) is the total number of edges between nodes whose degrees are equal to d 1 and d 2 respectively. We also find an asymptotic formula for the expectation of X(d 1, d 2, t) and prove a tight concentration result. Again, we do not impose any substantial restrictions on the values of d 1, d 2, and t.","PeriodicalId":38105,"journal":{"name":"Internet Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2011-08-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1080/15427951.2011.646176","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"59946845","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2011-06-14DOI: 10.1080/15427951.2011.566458
Angsheng Li, Pan Peng
Abstract Communities (or clusters) are ubiquitous in real-world networks. Researchers from different fields have proposed many definitions of communities, which are usually thought of as a subset of nodes whose vertices are well connected with other vertices in the set and have relatively fewer connections with vertices outside the set. In contrast to traditional research that focuses mainly on detecting and/or testing such clusters, we propose a new definition of community and a novel way to study community structure, with which we are able to investigate mathematical network models to test whether they exhibit the small-community phenomenon, i.e., whether every vertex in the network belongs to some small community. We examine various models and establish both positive and negative results: we show that in some models, the small-community phenomenon exists, while in some other models, it does not.
{"title":"Community Structures in Classical Network Models","authors":"Angsheng Li, Pan Peng","doi":"10.1080/15427951.2011.566458","DOIUrl":"https://doi.org/10.1080/15427951.2011.566458","url":null,"abstract":"Abstract Communities (or clusters) are ubiquitous in real-world networks. Researchers from different fields have proposed many definitions of communities, which are usually thought of as a subset of nodes whose vertices are well connected with other vertices in the set and have relatively fewer connections with vertices outside the set. In contrast to traditional research that focuses mainly on detecting and/or testing such clusters, we propose a new definition of community and a novel way to study community structure, with which we are able to investigate mathematical network models to test whether they exhibit the small-community phenomenon, i.e., whether every vertex in the network belongs to some small community. We examine various models and establish both positive and negative results: we show that in some models, the small-community phenomenon exists, while in some other models, it does not.","PeriodicalId":38105,"journal":{"name":"Internet Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2011-06-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1080/15427951.2011.566458","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"59946510","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2011-06-08DOI: 10.1080/15427951.2012.680824
D. Gleich, A. Owen
Abstract Stochastic Kronecker graphs supply a parsimonious model for large sparse real-world graphs. They can specify the distribution of a large random graph using only three or four parameters. Those parameters have, however, proved difficult to choose in specific applications. This article looks at method-of-moments estimators that are computationally much simpler than maximum likelihood. The estimators are fast, and in our examples, they typically yield Kronecker parameters with expected feature counts closer to a given graph than we get from KronFit. The improvement is especially prominent for the number of triangles in the graph.
{"title":"Moment-Based Estimation of Stochastic Kronecker Graph Parameters","authors":"D. Gleich, A. Owen","doi":"10.1080/15427951.2012.680824","DOIUrl":"https://doi.org/10.1080/15427951.2012.680824","url":null,"abstract":"Abstract Stochastic Kronecker graphs supply a parsimonious model for large sparse real-world graphs. They can specify the distribution of a large random graph using only three or four parameters. Those parameters have, however, proved difficult to choose in specific applications. This article looks at method-of-moments estimators that are computationally much simpler than maximum likelihood. The estimators are fast, and in our examples, they typically yield Kronecker parameters with expected feature counts closer to a given graph than we get from KronFit. The improvement is especially prominent for the number of triangles in the graph.","PeriodicalId":38105,"journal":{"name":"Internet Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2011-06-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1080/15427951.2012.680824","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"59947002","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2011-04-19DOI: 10.1080/15427951.2012.625256
F. Bonchi, Pooya Esfandiar, D. Gleich, C. Greif, L. Lakshmanan
Abstract We explore methods for approximating the commute time and Katz score between a pair of nodes. These methods are based on the approach of matrices, moments, and quadrature developed in the numerical linear algebra community. They rely on the Lanczos process and provide upper and lower bounds on an estimate of the pairwise scores. We also explore methods to approximate the commute times and Katz scores from a node to all other nodes in the graph. Here, our approach for the commute times is based on a variation of the conjugate gradient algorithm, and it provides an estimate of all the diagonals of the inverse of a matrix. Our technique for the Katz scores is based on exploiting an empirical localization property of the Katz matrix. We adapt algorithms used for personalized PageRank computing to these Katz scores and theoretically show that this approach is convergent. We evaluate these methods on 17 real-world graphs ranging in size from 1000 to 1,000,000 nodes. Our results show that our pairwise commute-time method and columnwise Katz algorithm both have attractive theoretical properties and empirical performance.
{"title":"Fast Matrix Computations for Pairwise and Columnwise Commute Times and Katz Scores","authors":"F. Bonchi, Pooya Esfandiar, D. Gleich, C. Greif, L. Lakshmanan","doi":"10.1080/15427951.2012.625256","DOIUrl":"https://doi.org/10.1080/15427951.2012.625256","url":null,"abstract":"Abstract We explore methods for approximating the commute time and Katz score between a pair of nodes. These methods are based on the approach of matrices, moments, and quadrature developed in the numerical linear algebra community. They rely on the Lanczos process and provide upper and lower bounds on an estimate of the pairwise scores. We also explore methods to approximate the commute times and Katz scores from a node to all other nodes in the graph. Here, our approach for the commute times is based on a variation of the conjugate gradient algorithm, and it provides an estimate of all the diagonals of the inverse of a matrix. Our technique for the Katz scores is based on exploiting an empirical localization property of the Katz matrix. We adapt algorithms used for personalized PageRank computing to these Katz scores and theoretically show that this approach is convergent. We evaluate these methods on 17 real-world graphs ranging in size from 1000 to 1,000,000 nodes. Our results show that our pairwise commute-time method and columnwise Katz algorithm both have attractive theoretical properties and empirical performance.","PeriodicalId":38105,"journal":{"name":"Internet Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2011-04-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1080/15427951.2012.625256","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"59946457","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2011-03-30DOI: 10.1080/15427951.2011.579849
Simla Ceyhan, M. Mousavi, A. Saberi
Abstract We propose a model for the evolution of market share in the presence of social influence. We study a simple market in which the individuals arrive sequentially and choose one of a number of available products. Their choice of product is a stochastic function of the inherent quality of the product and its market share. Using techniques from stochastic approximation theory, we show that market shares converge to an equilibrium. We also derive the market shares at equilibrium in terms of the level of social influence and the inherent quality of the products. In a special case, in which the choice model is a multinomial logit model, we show that inequality in the market increases with social influence and that with strong enough social influence, monopoly occurs. These results support the observations made in the experimental study of cultural markets in [Salganik et al. 06].
摘要本文提出了一个存在社会影响的市场份额演化模型。我们研究了一个简单的市场，在这个市场中，个人依次到达并从众多可用产品中选择一种。他们对产品的选择是产品内在质量及其市场份额的随机函数。利用随机逼近理论的技术，我们证明了市场份额收敛于一个均衡。我们还根据社会影响水平和产品的内在质量推导出均衡时的市场份额。在选择模型为多项式逻辑模型的特殊情况下，我们证明了市场中的不平等随着社会影响的增加而增加，当社会影响足够强时，就会出现垄断。这些结果支持了[Salganik et al. 06]在文化市场实验研究中的观察结果。
{"title":"Social Influence and Evolution of Market Share","authors":"Simla Ceyhan, M. Mousavi, A. Saberi","doi":"10.1080/15427951.2011.579849","DOIUrl":"https://doi.org/10.1080/15427951.2011.579849","url":null,"abstract":"Abstract We propose a model for the evolution of market share in the presence of social influence. We study a simple market in which the individuals arrive sequentially and choose one of a number of available products. Their choice of product is a stochastic function of the inherent quality of the product and its market share. Using techniques from stochastic approximation theory, we show that market shares converge to an equilibrium. We also derive the market shares at equilibrium in terms of the level of social influence and the inherent quality of the products. In a special case, in which the choice model is a multinomial logit model, we show that inequality in the market increases with social influence and that with strong enough social influence, monopoly occurs. These results support the observations made in the experimental study of cultural markets in [Salganik et al. 06].","PeriodicalId":38105,"journal":{"name":"Internet Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2011-03-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1080/15427951.2011.579849","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"59946576","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Online social networks have become very popular in recent years and their number of users is already measured in many hundreds of millions. For various commercial and sociological purposes, an independent estimate of their sizes is important. In this work, algorithms for estimating the number of users in such networks are considered. The proposed schemes are also applicable for estimating the sizes of networks' sub-populations. The suggested algorithms interact with the social networks via their public APIs only, and rely on no other external information. Due to obvious traffic and privacy concerns, the number of such interactions is severely limited. We therefore focus on minimizing the number of API interactions needed for producing good size estimates. We adopt the abstraction of social networks as undirected graphs and use random node sampling. By counting the number of collisions or non-unique nodes in the sample, we produce a size estimate. Then, we show analytically that the estimate error vanishes with high probability for smaller number of samples than those required by prior-art algorithms. Moreover, although our algorithms are provably correct for any graph, they excel when applied to social network-like graphs. The proposed algorithms were evaluated on synthetic as well real social networks such as Facebook, IMDB, and DBLP. Our experiments corroborated the theoretical results, and demonstrated the effectiveness of the algorithms.
{"title":"Estimating sizes of social networks via biased sampling","authors":"L. Katzir, Edo Liberty, O. Somekh, Ioana A. Cosma","doi":"10.1145/1963405.1963489","DOIUrl":"https://doi.org/10.1145/1963405.1963489","url":null,"abstract":"Online social networks have become very popular in recent years and their number of users is already measured in many hundreds of millions. For various commercial and sociological purposes, an independent estimate of their sizes is important. In this work, algorithms for estimating the number of users in such networks are considered. The proposed schemes are also applicable for estimating the sizes of networks' sub-populations. The suggested algorithms interact with the social networks via their public APIs only, and rely on no other external information. Due to obvious traffic and privacy concerns, the number of such interactions is severely limited. We therefore focus on minimizing the number of API interactions needed for producing good size estimates. We adopt the abstraction of social networks as undirected graphs and use random node sampling. By counting the number of collisions or non-unique nodes in the sample, we produce a size estimate. Then, we show analytically that the estimate error vanishes with high probability for smaller number of samples than those required by prior-art algorithms. Moreover, although our algorithms are provably correct for any graph, they excel when applied to social network-like graphs. The proposed algorithms were evaluated on synthetic as well real social networks such as Facebook, IMDB, and DBLP. Our experiments corroborated the theoretical results, and demonstrated the effectiveness of the algorithms.","PeriodicalId":38105,"journal":{"name":"Internet Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2011-03-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1145/1963405.1963489","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"64123853","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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